IODS course project
Donggyu Kam
About the project
I am so excited to get an opportunity to be into R. I am expecting to learn how to hand data and visulaze the results of analysis through R. One of my colleagues had let me know about this course.
The URL for my git repository as follow:
https://github.com/Donggyu-Kam/IODS-project
Regression and Model validation (Excercise 2)
The data is the student’s attitude toward statistics.The data consists of information about students’ ages, students’ gender, three approaches, students’ attitude toward statistics and exam points. The oberservations whare the exam points are zeor were removed from data. And thus, the data has 166 observations and 7 variables. You can figure out three variables to affect the exam points through linear regression models and finally you will figure out the most significant variable to exampoints
#Before handling data, you should install two packages:“ggplot2” and“GGally”.
1.To read and explore data
- Read data and look into the structure and dimension of data
- df<-read.table(“http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/learning2014.txt”, sep=“,”, header=TRUE )
- str(df)
- dim(df)
- Initialize plot with data and aesthetic mapping
- p1 <- ggplot(df, aes(x = attitude,col=gender, y = points))
- Define the visualization type (points)
- p2 <- p1 + geom_point()
- Draw the plot
- p2
- Add a regression line
- p3 <- p2 + geom_smooth(method = “lm”)
- Add a main title and draw the plot
- p4 <- p3+ggtitle(“<Student’s attitude versus exam points>”) &
- p4
2. To choose variables and create a regression model
By drawing plots illustrating the relationships between the variables and the normality of the erros distributions, it will find out what variables are closely related to exam points.
- Draw a scatter plot matrix of the variables in learning2014
- p <- ggpairs(df, mapping = aes(col=gender, alpha = 0.3), lower = list(combo = wrap(“facethist”, bins = 20)))
- Plotting
- p
According to the plot “p”, three variables “attitude”, “stra” and “surf” were chosen due to the strongest correlation value as explanatory variables.
- Fit a linear model to each variable
- qplot(attitude, points, data = df) + geom_smooth(method = “lm”)
- qplot(stra, points, data=df)+ geom_smooth(method =“lm”)
- qplot(surf, points, data=df)+ geom_smooth(method=“lm”)
After fitting each variable to a linear regression modle, “attitude” shows postive relationship but the others show week or negative relationships. So, the linear regression model is re-fitted with three variables together.
- Fit a linear modewl with three variables together
- mymodel<-lm(points~attitude+stra+surf, data=df)
- Take a look at the summary of the model
- summary(mymodel)
In this model, “Stra” and “Surf” are not not significantly related to the exam points. So, the model is re-fitted again witout the two variables
Fit the model again with the “attitude”
- mymodel2<- lm(points ~ attitude, data=df)
- summary(mymodel2)
- par(mfrow = c(2,2))
- plot(mymodel2, which=c(1,2,5)) - only take looks at the relation between residuals and fitted values, the normal QQplot, and the leverage of observations.
3. Result of regression model analysis
By analyzing the residuals and errors of the model, we can test the model validation. The variance of errors in residulas over fitted values looks constant without any patterns but there are 4 outliers indicating that the model only with “attitude” cannot explain the 4 ouliters of exam points. In the normality of errors in QQ plot, the errors are normaly distributed. In leverage of observations, standardized residuals are scattered around 0 and there seems no impact of outliers
Losistic Regression (Excercise 3)
The data is about student acheivement in secondary education of two Portugues schools. Two datasets are available for the performance in two subjects: Math and Portuguese language.
P. Cortez and A. Silva. Using Data Mining to Predict Secondary School Student Performance. In A. Brito and J. Teixeira Eds., Proceedings of 5th FUture BUsiness TEChnology Conference (FUBUTEC 2008) pp. 5-12, Porto, Portugal, April, 2008, EUROSIS, ISBN 978-9077381-39-7.[link]http://www3.dsi.uminho.pt/pcortez/student.pdf
1. Retreiving data
the data is about student acheivement in secondary education of two Portugues schools according to the student grades, demographic and school-related features. Two datasets are available for the performance in two subjects: Math and Portuguese language. the datasets consists of 35 variables and 382 observations.I firstly beging with installing some packages helps analysis. And thus, we can take a glimpse at the datasets and what variables are available through column names on the datasets.
library(dplyr);library(ggplot2);library(tidyr);library(boot)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionalc<-read.csv("Z:\\IODS-project\\data\\alc.csv")
glimpse(alc)## Observations: 382
## Variables: 36
## $ X <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
## $ school <fct> GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP,...
## $ sex <fct> F, F, F, F, F, M, M, F, M, M, F, F, M, M, M, F, F, ...
## $ age <int> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15,...
## $ address <fct> U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, ...
## $ famsize <fct> GT3, GT3, LE3, GT3, GT3, LE3, LE3, GT3, LE3, GT3, G...
## $ Pstatus <fct> A, T, T, T, T, T, T, A, A, T, T, T, T, T, A, T, T, ...
## $ Medu <int> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, ...
## $ Fedu <int> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, ...
## $ Mjob <fct> at_home, at_home, at_home, health, other, services,...
## $ Fjob <fct> teacher, other, other, services, other, other, othe...
## $ reason <fct> course, course, other, home, home, reputation, home...
## $ nursery <fct> yes, no, yes, yes, yes, yes, yes, yes, yes, yes, ye...
## $ internet <fct> no, yes, yes, yes, no, yes, yes, no, yes, yes, yes,...
## $ guardian <fct> mother, father, mother, mother, father, mother, mot...
## $ traveltime <int> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, ...
## $ studytime <int> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, ...
## $ failures <int> 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ schoolsup <fct> yes, no, yes, no, no, no, no, yes, no, no, no, no, ...
## $ famsup <fct> no, yes, no, yes, yes, yes, no, yes, yes, yes, yes,...
## $ paid <fct> no, no, yes, yes, yes, yes, no, no, yes, yes, yes, ...
## $ activities <fct> no, no, no, yes, no, yes, no, no, no, yes, no, yes,...
## $ higher <fct> yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, y...
## $ romantic <fct> no, no, no, yes, no, no, no, no, no, no, no, no, no...
## $ famrel <int> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, ...
## $ freetime <int> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, ...
## $ goout <int> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, ...
## $ Dalc <int> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
## $ Walc <int> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, ...
## $ health <int> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, ...
## $ absences <int> 6, 4, 10, 2, 4, 10, 0, 6, 0, 0, 0, 4, 2, 2, 0, 4, 6...
## $ G1 <int> 5, 5, 7, 15, 6, 15, 12, 6, 16, 14, 10, 10, 14, 10, ...
## $ G2 <int> 6, 5, 8, 14, 10, 15, 12, 5, 18, 15, 8, 12, 14, 10, ...
## $ G3 <int> 6, 6, 10, 15, 10, 15, 11, 6, 19, 15, 9, 12, 14, 11,...
## $ alc_use <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1...
## $ high_use <lgl> TRUE, TRUE, FALSE, TRUE, TRUE, TRUE, TRUE, TRUE, TR...gather(alc)%>% glimpse## Warning: attributes are not identical across measure variables;
## they will be dropped## Observations: 13,752
## Variables: 2
## $ key <chr> "X", "X", "X", "X", "X", "X", "X", "X", "X", "X", "X", "...
## $ value <chr> "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11",...gather(alc)%>%ggplot(aes(value))+facet_wrap("key", scales="free")+geom_bar()## Warning: attributes are not identical across measure variables;
## they will be dropped
2. Hypothesis
In order to study the relationship between high and low alcohol consumptions and variables, i am going to work folloing to hypothesis;
- Gender - Males consume more alcohol than females.
- G1 - First period grade decreases with alcohol consumption.
- G2 - Second period grade decreases with alcohol consumption.
- G3 - Third period grade decreases with alcohol consumption.
I will take a look at each relationship followring to the hypothesis one by one p<-ggpairs(alc, mapping = aes(col=sex, alpha = 0.3), lower = list(combo = wrap(“facethist”, bins = 20))) p
Hypothesis 1 - Gender
g1 <- ggplot(data = alc, aes(x = alc_use, fill = sex))
g1 + geom_bar()+facet_wrap("sex") + ggtitle("alohol consumption by sex")
g11<-ggplot(data=alc, aes(x=high_use, fill=sex))
g11+geom_bar()+facet_wrap("sex") + ggtitle("high and low alcohol consumption by sex")
Hypothesis 2 - G1
g2 <- ggplot(data=alc, aes(x=high_use, y= G1, col=sex))
g2+geom_boxplot()+ggtitle("G1 by alcohol consumption and sex")
Hypothesis 3 - G2
g3<-ggplot(alc, aes(x=high_use, y=G2, col=sex))
g3+geom_boxplot()+ggtitle("G2 by acohol consumption and sex")
Hypothesis 4 - Absences
g4<-ggplot(alc, aes(x=high_use, y=G3, col=sex))
g4+geom_boxplot()+ggtitle("G3 by alcohol consumption and sex")
Interim result
As a result of exploration of relationship with 4 variable to the high and low alcohol relationship, it seems that femailes consume more alcohol than males do. However, it is presumed that the more males consum alcohol, the less grades males achieve.
3.Lostistic regression
From now, in order to interpret relationship between variables presumably relating to alcohol consumptions and alcohol consumption, we will deal with logistaic regression.
m<-glm(high_use~sex+G1+G2+G3, data=alc, family="binomial")
summary(m)##
## Call:
## glm(formula = high_use ~ sex + G1 + G2 + G3, family = "binomial",
## data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.7050 -1.1393 0.8236 1.0275 1.4369
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.124551 0.377648 -0.330 0.742
## sexM -0.929752 0.215442 -4.316 1.59e-05 ***
## G1 0.100421 0.063556 1.580 0.114
## G2 0.005342 0.076244 0.070 0.944
## G3 -0.032884 0.053600 -0.614 0.540
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 524.94 on 381 degrees of freedom
## Residual deviance: 502.43 on 377 degrees of freedom
## AIC: 512.43
##
## Number of Fisher Scoring iterations: 4coef(m)## (Intercept) sexM G1 G2 G3
## -0.124550963 -0.929752246 0.100420779 0.005341903 -0.032884237OR<-coef(m)%>%exp
OR## (Intercept) sexM G1 G2 G3
## 0.8828933 0.3946515 1.1056360 1.0053562 0.9676506CI<-exp(confint(m))## Waiting for profiling to be done...CI## 2.5 % 97.5 %
## (Intercept) 0.4202206 1.8520181
## sexM 0.2575779 0.5999088
## G1 0.9774251 1.2553494
## G2 0.8649561 1.1681070
## G3 0.8686826 1.0737634cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.8828933 0.4202206 1.8520181
## sexM 0.3946515 0.2575779 0.5999088
## G1 1.1056360 0.9774251 1.2553494
## G2 1.0053562 0.8649561 1.1681070
## G3 0.9676506 0.8686826 1.0737634The model indicates taht males are more significantly related to high alcohol consumptions.However, the other 3 variables are not significantly related to high alcohol consumptions. In odds rates, the odds rates for G2 and G3 are more than 1.
Prediction power
probabilities<-predict(m, type="response")
alc<-mutate(alc,probability=probabilities)
alc<-mutate(alc,prediction=probability>0.5)
select(alc,G2,G3, high_use, probability, prediction)%>% tail(10)## G2 G3 high_use probability prediction
## 373 5 0 FALSE 0.3953044 FALSE
## 374 5 5 FALSE 0.3567518 FALSE
## 375 9 10 FALSE 0.6454027 TRUE
## 376 5 6 FALSE 0.5762451 TRUE
## 377 5 0 TRUE 0.6468232 TRUE
## 378 9 8 TRUE 0.5898627 TRUE
## 379 5 0 TRUE 0.6235593 TRUE
## 380 5 0 TRUE 0.6235593 TRUE
## 381 16 16 FALSE 0.4777423 FALSE
## 382 12 10 FALSE 0.4466009 FALSE2x2 cross-tabulation and graphic representation
g<-ggplot(alc,(aes(x=probability, y=high_use, col=prediction)))
g+geom_point()
table(high_use=alc$high_use, prediction=alc$prediction)%>%prop.table()%>%addmargins()## prediction
## high_use FALSE TRUE Sum
## FALSE 0.2225131 0.2225131 0.4450262
## TRUE 0.1492147 0.4057592 0.5549738
## Sum 0.3717277 0.6282723 1.0000000Exploring the training error
loss_func<-function(class, prob){
n_wrong <-abs(class - prob) > 0.5
mean(n_wrong)
}
loss_func(class = alc$high_use, prob = alc$probability)## [1] 0.3717277Through exploring the training error, the incorrection of prediction is almost 50%.
10-fold cross-validation (Bonus task)
cv<-cv.glm(data=alc, cost=loss_func, glmfit = m, K=10)
cv$delta[1]## [1] 0.395288Eventually, the prediction error with the 4 variables is 40%.
Clustering and classification (Exercise 4)
library(MASS);library(tidyverse);library(corrplot)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## select## -- Attaching packages --------------------------------------- tidyverse 1.2.1 --## v tibble 2.1.3 v purrr 0.3.3
## v readr 1.3.1 v stringr 1.4.0
## v tibble 2.1.3 v forcats 0.4.0## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()
## x MASS::select() masks dplyr::select()## corrplot 0.84 loadeddata("Boston")The data has 506 observations and 14 variables. Data is about housing values in suburbs of Boston.
Explort dataset
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00pairs(Boston)
Correlate all variables
cor_matrix<-cor(Boston)
cor_matrix%>%round(2)## crim zn indus chas nox rm age dis rad tax
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47
## ptratio black lstat medv
## crim 0.29 -0.39 0.46 -0.39
## zn -0.39 0.18 -0.41 0.36
## indus 0.38 -0.36 0.60 -0.48
## chas -0.12 0.05 -0.05 0.18
## nox 0.19 -0.38 0.59 -0.43
## rm -0.36 0.13 -0.61 0.70
## age 0.26 -0.27 0.60 -0.38
## dis -0.23 0.29 -0.50 0.25
## rad 0.46 -0.44 0.49 -0.38
## tax 0.46 -0.44 0.54 -0.47
## ptratio 1.00 -0.18 0.37 -0.51
## black -0.18 1.00 -0.37 0.33
## lstat 0.37 -0.37 1.00 -0.74
## medv -0.51 0.33 -0.74 1.00corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex=0.6)
There is strongly positive correlation bewtween “rad” and “tax” and stongly negative correlation between “age” and “dis” and between “lstat” and “medv”.
Standardized data
boston_scaled<-scale(Boston)
summary(boston_scaled)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865Covert matrix data to dafaframe
class(boston_scaled)## [1] "matrix"boston_scaled<-as.data.frame(boston_scaled)Create a catergorical variable with the variable “Crime”
summary("crim")## Length Class Mode
## 1 character characterbins<-quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610crime<-cut(boston_scaled$crim, breaks=bins, include.lowest=TRUE, label=c("low","med_low","med_high", "high"))
crime## [1] low low low low low low med_low
## [8] med_low med_low med_low med_low med_low med_low med_high
## [15] med_high med_high med_high med_high med_high med_high med_high
## [22] med_high med_high med_high med_high med_high med_high med_high
## [29] med_high med_high med_high med_high med_high med_high med_high
## [36] low med_low low med_low low low med_low
## [43] med_low med_low med_low med_low med_low med_low med_low
## [50] med_low med_low low low low low low
## [57] low low med_low med_low med_low med_low med_low
## [64] med_low low low low low med_low med_low
## [71] med_low med_low med_low med_low low med_low med_low
## [78] med_low low med_low low low low low
## [85] low low low low low low low
## [92] low low low low med_low med_low med_low
## [99] low low med_low med_low med_low med_low med_low
## [106] med_low med_low med_low med_low med_high med_low med_low
## [113] med_low med_low med_low med_low med_low med_low med_low
## [120] med_low low low med_low med_low med_low med_low
## [127] med_high med_high med_high med_high med_high med_high med_high
## [134] med_high med_high med_high med_high med_high med_low med_high
## [141] med_high med_high med_high high med_high med_high med_high
## [148] med_high med_high med_high med_high med_high med_high med_high
## [155] med_high med_high med_high med_high med_high med_high med_high
## [162] med_high med_high med_high med_high med_high med_high med_high
## [169] med_high med_high med_high med_high med_low med_low med_low
## [176] low low low low low low low
## [183] med_low med_low med_low low low low med_low
## [190] med_low med_low low med_low low low low
## [197] low low low low low low low
## [204] low low med_low med_low med_low med_low med_high
## [211] med_low med_high med_low med_low med_high med_low low
## [218] low med_low med_low med_high med_high med_high med_high
## [225] med_high med_high med_high med_high med_high med_high med_high
## [232] med_high med_high med_high med_high med_high med_high med_high
## [239] med_low med_low med_low med_low med_low med_low med_low
## [246] med_low med_high med_low med_low med_low med_low med_low
## [253] med_low med_high low low low med_high med_high
## [260] med_high med_high med_high med_high med_high med_high med_high
## [267] med_high med_high med_high med_low med_high med_low med_low
## [274] med_low low med_low med_low low low med_low
## [281] low low low low low low low
## [288] low low low low low low med_low
## [295] low med_low low med_low low low low
## [302] low med_low med_low low low low low
## [309] med_high med_high med_high med_high med_high med_high med_high
## [316] med_low med_high med_low med_high med_high med_low med_low
## [323] med_high med_high med_high med_low med_high med_low low
## [330] low low low low low low low
## [337] low low low low low low low
## [344] low low low low low low low
## [351] low low low low low med_low high
## [358] high high high high high high high
## [365] med_high high high high high high high
## [372] high high high high high high high
## [379] high high high high high high high
## [386] high high high high high high high
## [393] high high high high high high high
## [400] high high high high high high high
## [407] high high high high high high high
## [414] high high high high high high high
## [421] high high high high high high high
## [428] high high high high high high high
## [435] high high high high high high high
## [442] high high high high high high high
## [449] high high high high high high high
## [456] high high high high high high high
## [463] high high high med_high high high high
## [470] high high high med_high high high high
## [477] high high high high high high high
## [484] med_high med_high med_high high high med_low med_low
## [491] med_low med_low med_low med_low med_high med_low med_high
## [498] med_high med_low med_low med_low low low low
## [505] med_low low
## Levels: low med_low med_high hightable(crime)## crime
## low med_low med_high high
## 127 126 126 127Remove original “crim” from the dataset and add the new categorial variable
boston_scaled<-dplyr::select(boston_scaled,-crim)
boston_scaled## zn indus chas nox rm
## 1 0.28454827 -1.28663623 -0.2723291 -0.14407485 0.413262920
## 2 -0.48724019 -0.59279438 -0.2723291 -0.73953036 0.194082387
## 3 -0.48724019 -0.59279438 -0.2723291 -0.73953036 1.281445551
## 4 -0.48724019 -1.30558569 -0.2723291 -0.83445805 1.015297761
## 5 -0.48724019 -1.30558569 -0.2723291 -0.83445805 1.227362043
## 6 -0.48724019 -1.30558569 -0.2723291 -0.83445805 0.206891639
## 7 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.388026950
## 8 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.160306916
## 9 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.930285282
## 10 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.399412952
## 11 0.04872402 -0.47618230 -0.2723291 -0.26489191 0.131459378
## 12 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.392296701
## 13 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.563086727
## 14 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.477691714
## 15 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 16 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.641365488
## 17 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.497617217
## 18 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.419338455
## 19 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.179354069
## 20 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.793653261
## 21 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.017103545
## 22 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.454919710
## 23 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.203004422
## 24 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 25 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.513272969
## 26 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.975829289
## 27 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 28 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.338213193
## 29 -0.48724019 -0.43682573 -0.2723291 -0.14407485 0.299402903
## 30 -0.48724019 -0.43682573 -0.2723291 -0.14407485 0.554164692
## 31 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.813578764
## 32 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.302631937
## 33 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.476268464
## 34 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.830657767
## 35 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 36 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.500463717
## 37 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.631402737
## 38 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.618593485
## 39 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.453496460
## 40 2.72854505 -1.19334657 -0.2723291 -1.09335175 0.441727925
## 41 2.72854505 -1.19334657 -0.2723291 -1.09335175 1.052302267
## 42 -0.48724019 -0.61611679 -0.2723291 -0.92075595 0.690796712
## 43 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.164576667
## 44 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.104800158
## 45 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.306901688
## 46 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.857699521
## 47 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.709681499
## 48 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.362408446
## 49 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -1.260479332
## 50 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.971559538
## 51 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.457766211
## 52 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.241432178
## 53 0.41317968 -0.80123846 -0.2723291 -0.99842406 0.322174907
## 54 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.407952453
## 55 2.72854505 -1.04029322 -0.2723291 -1.24868797 -0.564509977
## 56 3.37170210 -1.44552019 -0.2723291 -1.30909650 1.372533565
## 57 3.15731641 -1.51548743 -0.2723291 -1.24868797 0.139998879
## 58 3.80047346 -1.43094368 -0.2723291 -1.24005818 0.756266222
## 59 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.198734672
## 60 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.509003218
## 61 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.773727758
## 62 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.453496460
## 63 0.58468822 -0.87557866 -0.2723291 -0.87760700 0.243896145
## 64 0.58468822 -0.87557866 -0.2723291 -0.87760700 0.679410711
## 65 0.26310970 -1.42219777 -0.2723291 -1.19604625 1.166162284
## 66 2.94293073 -1.13212523 -0.2723291 -1.35224545 0.007636609
## 67 2.94293073 -1.13212523 -0.2723291 -1.35224545 -0.708258248
## 68 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.578742479
## 69 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.982945540
## 70 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.568779727
## 71 -0.48724019 -0.04763292 -0.2723291 -1.22279860 0.188389387
## 72 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.460612711
## 73 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.312594689
## 74 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.056409650
## 75 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.016558644
## 76 -0.48724019 0.24681257 -0.2723291 -1.01568364 0.001943608
## 77 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.008019143
## 78 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.205850923
## 79 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.074911903
## 80 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.584435480
## 81 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.629596953
## 82 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.475885930
## 83 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.024715612
## 84 0.58468822 -0.91493524 -0.2723291 -1.11061133 -0.167423167
## 85 -0.48724019 -0.96886832 -0.2723291 -0.91212616 0.148538381
## 86 -0.48724019 -0.96886832 -0.2723291 -0.91212616 0.491541682
## 87 -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.383757200
## 88 -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.232892677
## 89 -0.48724019 -1.12629463 -0.2723291 -0.56693456 1.028107013
## 90 -0.48724019 -1.12629463 -0.2723291 -0.56693456 1.130581029
## 91 -0.48724019 -1.12629463 -0.2723291 -0.56693456 0.188389387
## 92 -0.48724019 -1.12629463 -0.2723291 -0.56693456 0.171310384
## 93 0.71331963 0.56895343 -0.2723291 -0.78267931 0.223970642
## 94 0.71331963 0.56895343 -0.2723291 -0.78267931 -0.104800158
## 95 0.71331963 0.56895343 -0.2723291 -0.78267931 -0.050716649
## 96 -0.48724019 -1.20209248 -0.2723291 -0.94664532 0.484425431
## 97 -0.48724019 -1.20209248 -0.2723291 -0.94664532 -0.173116168
## 98 -0.48724019 -1.20209248 -0.2723291 -0.94664532 2.539598741
## 99 -0.48724019 -1.20209248 -0.2723291 -0.94664532 2.185209437
## 100 -0.48724019 -1.20209248 -0.2723291 -0.94664532 1.610216351
## 101 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.629596953
## 102 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.706452465
## 103 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.171310384
## 104 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.210120673
## 105 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.167423167
## 106 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.617170235
## 107 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.638518988
## 108 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.224353176
## 109 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.269514649
## 110 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.079181654
## 111 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.127572161
## 112 -0.48724019 -0.16424500 -0.2723291 -0.06640675 0.612517950
## 113 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.528928721
## 114 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.274166933
## 115 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.043600398
## 116 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.507579968
## 117 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.154613915
## 118 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.375217698
## 119 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.587281980
## 120 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.787960260
## 121 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.590128481
## 122 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.399412952
## 123 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.460612711
## 124 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.610053984
## 125 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.577319229
## 126 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.425031456
## 127 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.955903786
## 128 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.842043769
## 129 -0.48724019 1.56744433 -0.2723291 0.59808708 0.208314890
## 130 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.921745781
## 131 -0.48724019 1.56744433 -0.2723291 0.59808708 0.246742645
## 132 -0.48724019 1.56744433 -0.2723291 0.59808708 0.058873617
## 133 -0.48724019 1.56744433 -0.2723291 0.59808708 0.124343127
## 134 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.658444491
## 135 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.750955755
## 136 -0.48724019 1.56744433 -0.2723291 0.59808708 0.071682869
## 137 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.487654465
## 138 -0.48724019 1.56744433 -0.2723291 0.59808708 0.241049645
## 139 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.608630733
## 140 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.190195170
## 141 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.157460416
## 142 -0.48724019 1.56744433 -0.2723291 0.59808708 -1.801314413
## 143 -0.48724019 1.23072696 3.6647712 2.72964520 -1.254786331
## 144 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.162275067
## 145 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.966411438
## 146 -0.48724019 1.23072696 -0.2723291 2.72964520 -0.220083425
## 147 -0.48724019 1.23072696 -0.2723291 2.72964520 -0.934555033
## 148 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.933676683
## 149 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.563631627
## 150 -0.48724019 1.23072696 -0.2723291 2.72964520 -0.978675789
## 151 -0.48724019 1.23072696 -0.2723291 2.72964520 -0.231469427
## 152 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.253363081
## 153 -0.48724019 1.23072696 3.6647712 2.72964520 -1.811277165
## 154 -0.48724019 1.23072696 -0.2723291 2.72964520 -0.819271765
## 155 -0.48724019 1.23072696 3.6647712 2.72964520 -0.221506675
## 156 -0.48724019 1.23072696 3.6647712 2.72964520 -0.188771920
## 157 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.441232109
## 158 -0.48724019 1.23072696 -0.2723291 0.43412107 0.937018999
## 159 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.311171439
## 160 -0.48724019 1.23072696 -0.2723291 2.72964520 0.320751657
## 161 -0.48724019 1.23072696 3.6647712 0.43412107 -0.049293399
## 162 -0.48724019 1.23072696 -0.2723291 0.43412107 1.714113616
## 163 -0.48724019 1.23072696 3.6647712 0.43412107 2.159590934
## 164 -0.48724019 1.23072696 3.6647712 0.43412107 2.975113306
## 165 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.612900484
## 166 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.261357681
## 167 -0.48724019 1.23072696 -0.2723291 0.43412107 2.340343711
## 168 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.580165729
## 169 -0.48724019 1.23072696 -0.2723291 0.43412107 0.048910866
## 170 -0.48724019 1.23072696 -0.2723291 0.43412107 0.167040633
## 171 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.583012230
## 172 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.575895979
## 173 -0.48724019 -1.03300497 -0.2723291 -0.38570897 -1.014257045
## 174 -0.48724019 -1.03300497 -0.2723291 -0.38570897 0.186966136
## 175 -0.48724019 -1.03300497 -0.2723291 -0.38570897 -0.605784233
## 176 -0.48724019 -1.03300497 -0.2723291 -0.38570897 0.371988664
## 177 -0.48724019 -1.03300497 -0.2723291 -0.38570897 -0.376640949
## 178 -0.48724019 -1.03300497 -0.2723291 -0.38570897 0.043217865
## 179 -0.48724019 -1.03300497 -0.2723291 -0.38570897 0.818889232
## 180 -0.48724019 -1.26477147 -0.2723291 -0.57556435 0.989679257
## 181 -0.48724019 -1.26477147 -0.2723291 -0.57556435 2.106930676
## 182 -0.48724019 -1.26477147 -0.2723291 -0.57556435 -0.200157922
## 183 -0.48724019 -1.26477147 -0.2723291 -0.57556435 1.238748045
## 184 -0.48724019 -1.26477147 -0.2723291 -0.57556435 0.396183918
## 185 -0.48724019 -1.26477147 -0.2723291 -0.57556435 -0.968713038
## 186 -0.48724019 -1.26477147 -0.2723291 -0.57556435 -0.187348670
## 187 -0.48724019 -1.26477147 -0.2723291 -0.57556435 2.200865190
## 188 1.44223095 -1.12192167 -0.2723291 -1.01568364 0.707875715
## 189 1.44223095 -1.12192167 -0.2723291 -1.01568364 0.386221166
## 190 1.44223095 -1.12192167 -0.2723291 -1.01568364 1.281445551
## 191 1.44223095 -1.12192167 -0.2723291 -1.01568364 0.948405001
## 192 1.44223095 -1.12192167 -0.2723291 -1.01568364 0.646675956
## 193 1.44223095 -1.12192167 -0.2723291 -1.01568364 1.271482800
## 194 2.08538800 -1.19626187 -0.2723291 -1.32635608 0.733494219
## 195 2.08538800 -1.19626187 -0.2723291 -1.32635608 0.454537177
## 196 2.94293073 -1.55630166 -0.2723291 -1.14513049 2.263488199
## 197 2.94293073 -1.40179066 -0.2723291 -1.30046671 1.426617073
## 198 2.94293073 -1.40179066 -0.2723291 -1.30046671 1.170432035
## 199 2.94293073 -1.40179066 -0.2723291 -1.30046671 1.408114820
## 200 3.58608778 -1.40907891 -0.2723291 -1.30909650 0.982563006
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## 436 -0.48724019 1.01499462 -0.2723291 1.59914271 0.490118432
## 437 -0.48724019 1.01499462 -0.2723291 1.59914271 0.251012396
## 438 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.188771920
## 439 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.497617217
## 440 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.935978283
## 441 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.664137492
## 442 -0.48724019 1.01499462 -0.2723291 1.59914271 0.172733634
## 443 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.093414156
## 444 -0.48724019 1.01499462 -0.2723291 1.59914271 0.285170401
## 445 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.612900484
## 446 -0.48724019 1.01499462 -0.2723291 1.59914271 0.248165896
## 447 -0.48724019 1.01499462 -0.2723291 1.59914271 0.080222370
## 448 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.047870149
## 449 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.141804663
## 450 -0.48724019 1.01499462 -0.2723291 1.36613839 0.188389387
## 451 -0.48724019 1.01499462 -0.2723291 1.36613839 0.660908458
## 452 -0.48724019 1.01499462 -0.2723291 1.36613839 0.527122938
## 453 -0.48724019 1.01499462 -0.2723291 1.36613839 0.017599361
## 454 -0.48724019 1.01499462 -0.2723291 1.36613839 1.577481596
## 455 -0.48724019 1.01499462 -0.2723291 1.36613839 0.631020203
## 456 -0.48724019 1.01499462 -0.2723291 1.36613839 0.342100410
## 457 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.439263958
## 458 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.496193967
## 459 -0.48724019 1.01499462 -0.2723291 1.36613839 0.023292362
## 460 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.289822685
## 461 -0.48724019 1.01499462 -0.2723291 1.36613839 0.592592447
## 462 -0.48724019 1.01499462 -0.2723291 1.36613839 0.130036128
## 463 -0.48724019 1.01499462 -0.2723291 1.36613839 0.046064365
## 464 -0.48724019 1.01499462 -0.2723291 1.36613839 0.325021407
## 465 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.107646658
## 466 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.748109254
## 467 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.473421963
## 468 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.400836202
## 469 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.510426469
## 470 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.813578764
## 471 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.167423167
## 472 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.079181654
## 473 -0.48724019 1.01499462 -0.2723291 0.21837632 0.216854391
## 474 -0.48724019 1.01499462 -0.2723291 0.51178918 0.989679257
## 475 -0.48724019 1.01499462 -0.2723291 0.25289548 -1.220628326
## 476 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.174539418
## 477 -0.48724019 1.01499462 -0.2723291 0.51178918 0.283747151
## 478 -0.48724019 1.01499462 -0.2723291 0.51178918 -1.395688102
## 479 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.141804663
## 480 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.079181654
## 481 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.060679401
## 482 -0.48724019 1.01499462 -0.2723291 -0.19585359 0.662331708
## 483 -0.48724019 1.01499462 -0.2723291 -0.19585359 1.104962525
## 484 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.743839504
## 485 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.588705230
## 486 -0.48724019 1.01499462 -0.2723291 0.24426569 0.038948114
## 487 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.242855428
## 488 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.540314723
## 489 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.182200570
## 490 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.239130578
## 491 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.695993897
## 492 -0.48724019 2.42017014 -0.2723291 0.46864023 -0.429301206
## 493 -0.48724019 2.42017014 -0.2723291 0.46864023 -0.429301206
## 494 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.822118266
## 495 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.510426469
## 496 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.874778524
## 497 -0.48724019 -0.21088983 -0.2723291 0.26152527 -1.273288584
## 498 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.698295497
## 499 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.378064199
## 500 -0.48724019 -0.21088983 -0.2723291 0.26152527 -1.018526795
## 501 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.366678197
## 502 -0.48724019 0.11562398 -0.2723291 0.15796779 0.438881424
## 503 -0.48724019 0.11562398 -0.2723291 0.15796779 -0.234315927
## 504 -0.48724019 0.11562398 -0.2723291 0.15796779 0.983986256
## 505 -0.48724019 0.11562398 -0.2723291 0.15796779 0.724954717
## 506 -0.48724019 0.11562398 -0.2723291 0.15796779 -0.362408446
## age dis rad tax ptratio
## 1 -0.119894767 0.1400749840 -0.9818712 -0.66594918 -1.45755797
## 2 0.366803426 0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 3 -0.265548971 0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 4 -0.809087830 1.0766711351 -0.7521778 -1.10502160 0.11292035
## 5 -0.510674339 1.0766711351 -0.7521778 -1.10502160 0.11292035
## 6 -0.350809969 1.0766711351 -0.7521778 -1.10502160 0.11292035
## 7 -0.070159185 0.8384142195 -0.5224844 -0.57694801 -1.50374851
## 8 0.977840575 1.0236248974 -0.5224844 -0.57694801 -1.50374851
## 9 1.116389695 1.0861216287 -0.5224844 -0.57694801 -1.50374851
## 10 0.615481336 1.3283202075 -0.5224844 -0.57694801 -1.50374851
## 11 0.913894827 1.2117799501 -0.5224844 -0.57694801 -1.50374851
## 12 0.508905089 1.1547920492 -0.5224844 -0.57694801 -1.50374851
## 13 -1.050660656 0.7863652700 -0.5224844 -0.57694801 -1.50374851
## 14 -0.240681180 0.4333252240 -0.6373311 -0.60068166 1.17530274
## 15 0.565745754 0.3166899868 -0.6373311 -0.60068166 1.17530274
## 16 -0.428965883 0.3341187865 -0.6373311 -0.60068166 1.17530274
## 17 -1.395257187 0.3341187865 -0.6373311 -0.60068166 1.17530274
## 18 0.466274590 0.2198105553 -0.6373311 -0.60068166 1.17530274
## 19 -1.135921653 0.0006920764 -0.6373311 -0.60068166 1.17530274
## 20 0.032864520 0.0006920764 -0.6373311 -0.60068166 1.17530274
## 21 1.048891406 0.0013569352 -0.6373311 -0.60068166 1.17530274
## 22 0.732715207 0.1031753182 -0.6373311 -0.60068166 1.17530274
## 23 0.821528746 0.0863638874 -0.6373311 -0.60068166 1.17530274
## 24 1.116389695 0.1425444597 -0.6373311 -0.60068166 1.17530274
## 25 0.906789743 0.2871037683 -0.6373311 -0.60068166 1.17530274
## 26 0.608376252 0.3132232229 -0.6373311 -0.60068166 1.17530274
## 27 0.771793164 0.4212152950 -0.6373311 -0.60068166 1.17530274
## 28 0.718505041 0.3126533438 -0.6373311 -0.60068166 1.17530274
## 29 0.917447368 0.3132707128 -0.6373311 -0.60068166 1.17530274
## 30 0.665216917 0.2108349609 -0.6373311 -0.60068166 1.17530274
## 31 0.906789743 0.2079855659 -0.6373311 -0.60068166 1.17530274
## 32 1.116389695 0.1804414138 -0.6373311 -0.60068166 1.17530274
## 33 0.476932215 0.0925850666 -0.6373311 -0.60068166 1.17530274
## 34 0.938762618 -0.0037244859 -0.6373311 -0.60068166 1.17530274
## 35 1.006260907 -0.0167367233 -0.6373311 -0.60068166 1.17530274
## 36 -0.013318520 -0.2064589434 -0.5224844 -0.76681717 0.34387304
## 37 -0.254891346 -0.1981007179 -0.5224844 -0.76681717 0.34387304
## 38 -0.961847117 0.0660856927 -0.5224844 -0.76681717 0.34387304
## 39 -1.363284313 0.0248169544 -0.5224844 -0.76681717 0.34387304
## 40 -1.661697804 0.7627152911 -0.7521778 -0.92701927 -0.07184181
## 41 -1.874850298 0.7627152911 -0.7521778 -0.92701927 -0.07184181
## 42 -2.333128159 0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 43 -2.201684121 0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 44 -2.205236663 0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 45 -1.015135240 0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 46 -1.235392817 0.6199131095 -0.7521778 -1.03975408 -0.25660396
## 47 -1.253155525 0.6199131095 -0.7521778 -1.03975408 -0.25660396
## 48 0.601271169 0.8996287230 -0.7521778 -1.03975408 -0.25660396
## 49 0.949420242 0.9853955139 -0.7521778 -1.03975408 -0.25660396
## 50 -0.233576097 1.0887810641 -0.7521778 -1.03975408 -0.25660396
## 51 -0.812640371 1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 52 -0.198050682 1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 53 -1.686565595 1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 54 -1.675907970 1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 55 -0.745142082 1.6738568465 -0.7521778 0.36053095 1.22149328
## 56 -1.658145262 2.3277455193 -0.5224844 -1.08128796 -0.25660396
## 57 -1.167894527 2.5609210138 -0.8670245 -0.56508119 -0.53374719
## 58 -0.997372532 2.1511780064 -0.5224844 -0.90328562 -1.54993904
## 59 -1.398809729 1.9089794276 -0.1779443 -0.73715011 0.57482574
## 60 -0.759352248 1.4897384367 -0.1779443 -0.73715011 0.57482574
## 61 -0.084369352 1.6290738544 -0.1779443 -0.73715011 0.57482574
## 62 0.881921953 1.4358373805 -0.1779443 -0.73715011 0.57482574
## 63 -0.027528687 1.6291213443 -0.1779443 -0.73715011 0.57482574
## 64 -0.894348827 1.9878601804 -0.1779443 -0.73715011 0.57482574
## 65 -0.322389636 2.5776849546 -0.7521778 -1.14062207 0.06672981
## 66 -1.803799466 1.3375332514 -0.6373311 -0.42267932 -1.08803366
## 67 -1.331311439 1.3375332514 -0.6373311 -0.42267932 -1.08803366
## 68 -1.675907970 1.2836321952 -0.6373311 -0.37521203 0.20530143
## 69 -1.128816570 1.2836321952 -0.6373311 -0.37521203 0.20530143
## 70 -1.263813149 1.2836321952 -0.6373311 -0.37521203 0.20530143
## 71 -2.201684121 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 72 -1.814457091 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 73 -2.159053622 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 74 -2.215894287 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 75 -2.222999370 0.2167712006 -0.5224844 -0.06074124 0.11292035
## 76 -0.837508162 0.3360183832 -0.5224844 -0.06074124 0.11292035
## 77 0.210491598 0.1221237952 -0.5224844 -0.06074124 0.11292035
## 78 -0.809087830 0.1403124336 -0.5224844 -0.06074124 0.11292035
## 79 -0.528437047 0.5789293108 -0.5224844 -0.06074124 0.11292035
## 80 -1.135921653 0.3360183832 -0.5224844 -0.06074124 0.11292035
## 81 -1.246050442 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 82 0.064837394 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 83 -1.292233482 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 84 -0.777114956 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 85 -0.730931915 0.4674704746 -0.7521778 -0.95668632 0.02053927
## 86 -0.443176049 0.3051974268 -0.7521778 -0.95668632 0.02053927
## 87 -0.833955621 0.3002109855 -0.7521778 -0.95668632 0.02053927
## 88 -0.418308258 -0.0225304932 -0.7521778 -0.95668632 0.02053927
## 89 0.629691502 -0.1773001341 -0.8670245 -0.82021787 -0.30279450
## 90 -0.194498140 -0.1807194081 -0.8670245 -0.82021787 -0.30279450
## 91 -0.087921893 -0.3337319220 -0.8670245 -0.82021787 -0.30279450
## 92 0.189176348 -0.3338269018 -0.8670245 -0.82021787 -0.30279450
## 93 -0.531989588 -0.0613297558 -0.6373311 -0.82021787 -0.11803234
## 94 -1.409467353 -0.0613297558 -0.6373311 -0.82021787 -0.11803234
## 95 0.309962761 -0.0855021237 -0.6373311 -0.82021787 -0.11803234
## 96 -0.382782843 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 97 0.036417061 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 98 0.263779721 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 99 -1.125264029 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 100 -0.215813389 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 101 0.402328842 -0.4830877123 -0.5224844 -0.14380900 1.12911220
## 102 0.096810268 -0.4459031069 -0.5224844 -0.14380900 1.12911220
## 103 0.597718628 -0.5130538501 -0.5224844 -0.14380900 1.12911220
## 104 0.668769459 -0.5130538501 -0.5224844 -0.14380900 1.12911220
## 105 0.761135540 -0.6525317376 -0.5224844 -0.14380900 1.12911220
## 106 0.999155824 -0.8016975682 -0.5224844 -0.14380900 1.12911220
## 107 0.828633829 -0.7522605641 -0.5224844 -0.14380900 1.12911220
## 108 0.590613545 -0.7943366310 -0.5224844 -0.14380900 1.12911220
## 109 1.013365990 -0.6468804374 -0.5224844 -0.14380900 1.12911220
## 110 0.803766038 -0.5935967501 -0.5224844 -0.14380900 1.12911220
## 111 -0.503569256 -0.4830877123 -0.5224844 -0.14380900 1.12911220
## 112 0.462722049 -0.5307200994 -0.4076377 0.14099473 -0.30279450
## 113 0.864159245 -0.6846349217 -0.4076377 0.14099473 -0.30279450
## 114 0.952972784 -0.5922195425 -0.4076377 0.14099473 -0.30279450
## 115 0.555088129 -0.7306526517 -0.4076377 0.14099473 -0.30279450
## 116 0.697189791 -0.6325384823 -0.4076377 0.14099473 -0.30279450
## 117 0.139440767 -0.5057404029 -0.4076377 0.14099473 -0.30279450
## 118 0.498247464 -0.4975246471 -0.4076377 0.14099473 -0.30279450
## 119 0.160756016 -0.6256999342 -0.4076377 0.14099473 -0.30279450
## 120 -0.119894767 -0.4919208369 -0.4076377 0.14099473 -0.30279450
## 121 0.039969603 -0.7300827727 -0.8670245 -1.30675758 0.29768250
## 122 0.551535588 -0.7587191929 -0.8670245 -1.30675758 0.29768250
## 123 0.864159245 -0.8111955516 -0.8670245 -1.30675758 0.29768250
## 124 1.009813449 -0.8788686839 -0.8670245 -1.30675758 0.29768250
## 125 0.967182950 -0.8494724251 -0.8670245 -1.30675758 0.29768250
## 126 0.704294875 -0.8558360740 -0.8670245 -1.30675758 0.29768250
## 127 0.960077867 -0.9677698093 -0.8670245 -1.30675758 0.29768250
## 128 0.974288033 -0.9530004450 -0.6373311 0.17066179 1.26768382
## 129 1.073759197 -0.9415078850 -0.6373311 0.17066179 1.26768382
## 130 0.928104993 -0.8620097633 -0.6373311 0.17066179 1.26768382
## 131 1.077311738 -0.7961887377 -0.6373311 0.17066179 1.26768382
## 132 1.034681240 -0.7237666137 -0.6373311 0.17066179 1.26768382
## 133 1.041786323 -0.6969823003 -0.6373311 0.17066179 1.26768382
## 134 0.952972784 -0.6293091680 -0.6373311 0.17066179 1.26768382
## 135 1.059549031 -0.6881491756 -0.6373311 0.17066179 1.26768382
## 136 1.052443947 -0.7998929513 -0.6373311 0.17066179 1.26768382
## 137 0.885474494 -0.8681834525 -0.6373311 0.17066179 1.26768382
## 138 1.059549031 -0.9237941458 -0.6373311 0.17066179 1.26768382
## 139 1.052443947 -1.0098458762 -0.6373311 0.17066179 1.26768382
## 140 1.041786323 -1.0097983862 -0.6373311 0.17066179 1.26768382
## 141 0.889027036 -1.0367726593 -0.6373311 0.17066179 1.26768382
## 142 1.116389695 -1.1186927669 -0.6373311 0.17066179 1.26768382
## 143 1.116389695 -1.1746358896 -0.5224844 -0.03107419 -1.73470120
## 144 1.116389695 -1.1317999841 -0.5224844 -0.03107419 -1.73470120
## 145 1.038233781 -1.1630958396 -0.5224844 -0.03107419 -1.73470120
## 146 1.116389695 -1.1283332201 -0.5224844 -0.03107419 -1.73470120
## 147 1.116389695 -1.0820305506 -0.5224844 -0.03107419 -1.73470120
## 148 0.963630408 -1.1085299245 -0.5224844 -0.03107419 -1.73470120
## 149 0.896132119 -1.0758568614 -0.5224844 -0.03107419 -1.73470120
## 150 0.935210076 -1.0777089681 -0.5224844 -0.03107419 -1.73470120
## 151 1.020471073 -1.0338757744 -0.5224844 -0.03107419 -1.73470120
## 152 1.116389695 -1.0464131126 -0.5224844 -0.03107419 -1.73470120
## 153 0.690084708 -1.0375799879 -0.5224844 -0.03107419 -1.73470120
## 154 1.063101572 -1.0314062987 -0.5224844 -0.03107419 -1.73470120
## 155 0.974288033 -0.9714740229 -0.5224844 -0.03107419 -1.73470120
## 156 0.498247464 -0.9733261297 -0.5224844 -0.03107419 -1.73470120
## 157 0.903237202 -0.9776477121 -0.5224844 -0.03107419 -1.73470120
## 158 1.024023615 -0.9107344185 -0.5224844 -0.03107419 -1.73470120
## 159 1.116389695 -0.9677223194 -0.5224844 -0.03107419 -1.73470120
## 160 1.116389695 -0.9636381865 -0.5224844 -0.03107419 -1.73470120
## 161 0.853501620 -0.9482039634 -0.5224844 -0.03107419 -1.73470120
## 162 0.789555872 -0.8662838558 -0.5224844 -0.03107419 -1.73470120
## 163 1.052443947 -0.8331358935 -0.5224844 -0.03107419 -1.73470120
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## 393 1.009813449 -0.9616910999 1.6596029 1.52941294 0.80577843
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## 395 0.928104993 -0.9559448199 1.6596029 1.52941294 0.80577843
## 396 1.073759197 -0.9827291333 1.6596029 1.52941294 0.80577843
## 397 0.974288033 -1.0059517029 1.6596029 1.52941294 0.80577843
## 398 1.077311738 -1.0265623271 1.6596029 1.52941294 0.80577843
## 399 1.116389695 -1.0948528283 1.6596029 1.52941294 0.80577843
## 400 0.327725469 -1.0897239172 1.6596029 1.52941294 0.80577843
## 401 1.116389695 -1.0477428302 1.6596029 1.52941294 0.80577843
## 402 1.116389695 -1.0547238481 1.6596029 1.52941294 0.80577843
## 403 1.116389695 -1.0239028917 1.6596029 1.52941294 0.80577843
## 404 0.974288033 -0.9936043244 1.6596029 1.52941294 0.80577843
## 405 0.597718628 -1.0389097056 1.6596029 1.52941294 0.80577843
## 406 1.116389695 -1.1253413553 1.6596029 1.52941294 0.80577843
## 407 1.116389695 -1.2427839210 1.6596029 1.52941294 0.80577843
## 408 1.116389695 -1.1919222195 1.6596029 1.52941294 0.80577843
## 409 1.041786323 -1.1114268095 1.6596029 1.52941294 0.80577843
## 410 1.116389695 -1.1062978984 1.6596029 1.52941294 0.80577843
## 411 1.116389695 -1.1312301050 1.6596029 1.52941294 0.80577843
## 412 1.116389695 -1.0768541496 1.6596029 1.52941294 0.80577843
## 413 1.116389695 -1.0643168114 1.6596029 1.52941294 0.80577843
## 414 1.116389695 -1.0474578907 1.6596029 1.52941294 0.80577843
## 415 1.116389695 -1.0147848276 1.6596029 1.52941294 0.80577843
## 416 1.116389695 -0.9309651233 1.6596029 1.52941294 0.80577843
## 417 0.789555872 -0.9381835908 1.6596029 1.52941294 0.80577843
## 418 0.729162665 -1.0198662487 1.6596029 1.52941294 0.80577843
## 419 1.116389695 -0.9462093868 1.6596029 1.52941294 0.80577843
## 420 0.281542429 -0.9502935197 1.6596029 1.52941294 0.80577843
## 421 1.116389695 -0.9194725633 1.6596029 1.52941294 0.80577843
## 422 0.949420242 -0.9120166463 1.6596029 1.52941294 0.80577843
## 423 0.675874542 -0.8756393696 1.6596029 1.52941294 0.80577843
## 424 0.587061003 -0.8421114879 1.6596029 1.52941294 0.80577843
## 425 0.071942477 -0.8223081923 1.6596029 1.52941294 0.80577843
## 426 0.952972784 -0.8953951752 1.6596029 1.52941294 0.80577843
## 427 -0.315284553 -0.8536040479 1.6596029 1.52941294 0.80577843
## 428 0.359698343 -0.9175729666 1.6596029 1.52941294 0.80577843
## 429 0.338383094 -0.8830477967 1.6596029 1.52941294 0.80577843
## 430 0.960077867 -0.8675660836 1.6596029 1.52941294 0.80577843
## 431 0.622586419 -0.8274371034 1.6596029 1.52941294 0.80577843
## 432 0.913894827 -0.8105781827 1.6596029 1.52941294 0.80577843
## 433 0.221149222 -0.7572944954 1.6596029 1.52941294 0.80577843
## 434 0.686532167 -0.7024911307 1.6596029 1.52941294 0.80577843
## 435 0.938762618 -0.7469416934 1.6596029 1.52941294 0.80577843
## 436 0.924552451 -0.7932443629 1.6596029 1.52941294 0.80577843
## 437 0.878369411 -0.8512295520 1.6596029 1.52941294 0.80577843
## 438 1.116389695 -0.8932106390 1.6596029 1.52941294 0.80577843
## 439 0.686532167 -0.9376612017 1.6596029 1.52941294 0.80577843
## 440 0.899684660 -0.9392758589 1.6596029 1.52941294 0.80577843
## 441 0.846396537 -0.9160057994 1.6596029 1.52941294 0.80577843
## 442 1.016918532 -0.8215483536 1.6596029 1.52941294 0.80577843
## 443 1.116389695 -0.8501847738 1.6596029 1.52941294 0.80577843
## 444 1.116389695 -0.8627221120 1.6596029 1.52941294 0.80577843
## 445 0.995603282 -0.9020437636 1.6596029 1.52941294 0.80577843
## 446 0.931657534 -0.8582105699 1.6596029 1.52941294 0.80577843
## 447 0.988498199 -0.8182715493 1.6596029 1.52941294 0.80577843
## 448 0.995603282 -0.7584342534 1.6596029 1.52941294 0.80577843
## 449 1.070206655 -0.7282306659 1.6596029 1.52941294 0.80577843
## 450 1.055996489 -0.7646079427 1.6596029 1.52941294 0.80577843
## 451 0.853501620 -0.6987869171 1.6596029 1.52941294 0.80577843
## 452 1.052443947 -0.6837801032 1.6596029 1.52941294 0.80577843
## 453 0.825081288 -0.6776064140 1.6596029 1.52941294 0.80577843
## 454 1.091521905 -0.6374774338 1.6596029 1.52941294 0.80577843
## 455 0.906789743 -0.6168668096 1.6596029 1.52941294 0.80577843
## 456 0.636796585 -0.6455032298 1.6596029 1.52941294 0.80577843
## 457 0.686532167 -0.5767378294 1.6596029 1.52941294 0.80577843
## 458 0.416539008 -0.4824228534 1.6596029 1.52941294 0.80577843
## 459 0.537325421 -0.4805707466 1.6596029 1.52941294 0.80577843
## 460 0.562193212 -0.5117241325 1.6596029 1.52941294 0.80577843
## 461 0.761135540 -0.5687120333 1.6596029 1.52941294 0.80577843
## 462 0.704294875 -0.5831489682 1.6596029 1.52941294 0.80577843
## 463 0.512457630 -0.5036983364 1.6596029 1.52941294 0.80577843
## 464 0.757582998 -0.4717851119 1.6596029 1.52941294 0.80577843
## 465 -0.112789684 -0.3949464255 1.6596029 1.52941294 0.80577843
## 466 -0.723826832 -0.3459843207 1.6596029 1.52941294 0.80577843
## 467 0.572850837 -0.4385896596 1.6596029 1.52941294 0.80577843
## 468 0.920999910 -0.5958762661 1.6596029 1.52941294 0.80577843
## 469 0.086152643 -0.4210658801 1.6596029 1.52941294 0.80577843
## 470 -0.421860800 -0.4612898402 1.6596029 1.52941294 0.80577843
## 471 0.547983046 -0.3617034834 1.6596029 1.52941294 0.80577843
## 472 0.786003330 -0.3304076278 1.6596029 1.52941294 0.80577843
## 473 0.228254306 -0.4267171803 1.6596029 1.52941294 0.80577843
## 474 -0.034633770 -0.5993905200 1.6596029 1.52941294 0.80577843
## 475 0.952972784 -0.6483526248 1.6596029 1.52941294 0.80577843
## 476 1.024023615 -0.7546350600 1.6596029 1.52941294 0.80577843
## 477 0.889027036 -0.7074775720 1.6596029 1.52941294 0.80577843
## 478 1.020471073 -0.8046419430 1.6596029 1.52941294 0.80577843
## 479 0.999155824 -0.7714939807 1.6596029 1.52941294 0.80577843
## 480 0.690084708 -0.8756393696 1.6596029 1.52941294 0.80577843
## 481 -0.137657475 -0.1761128861 1.6596029 1.52941294 0.80577843
## 482 0.224701764 -0.2200410597 1.6596029 1.52941294 0.80577843
## 483 0.299305137 -0.1825715149 1.6596029 1.52941294 0.80577843
## 484 -1.004477616 0.1440166471 1.6596029 1.52941294 0.80577843
## 485 -0.947636951 -0.0337381137 1.6596029 1.52941294 0.80577843
## 486 -0.592382795 0.0933923952 1.6596029 1.52941294 0.80577843
## 487 0.398776300 -0.1183176566 1.6596029 1.52941294 0.80577843
## 488 -0.546199754 -0.3052379716 1.6596029 1.52941294 0.80577843
## 489 0.857054162 -0.9375187319 -0.6373311 1.79641644 0.75958789
## 490 1.055996489 -0.9686246278 -0.6373311 1.79641644 0.75958789
## 491 1.045338864 -0.9367114033 -0.6373311 1.79641644 0.75958789
## 492 1.073759197 -0.9151034909 -0.6373311 1.79641644 0.75958789
## 493 0.530220338 -0.8002728706 -0.6373311 1.79641644 0.75958789
## 494 -0.517779422 -0.6711952751 -0.4076377 -0.10227512 0.34387304
## 495 -0.922769160 -0.6711952751 -0.4076377 -0.10227512 0.34387304
## 496 -1.413019895 -0.4732098094 -0.4076377 -0.10227512 0.34387304
## 497 0.153650933 -0.4732098094 -0.4076377 -0.10227512 0.34387304
## 498 0.071942477 -0.4285217972 -0.4076377 -0.10227512 0.34387304
## 499 -0.116342226 -0.6581830377 -0.4076377 -0.10227512 0.34387304
## 500 0.174966182 -0.6625521101 -0.4076377 -0.10227512 0.34387304
## 501 0.395223759 -0.6158695213 -0.4076377 -0.10227512 0.34387304
## 502 0.018654354 -0.6251775451 -0.9818712 -0.80241764 1.17530274
## 503 0.288647512 -0.7159307773 -0.9818712 -0.80241764 1.17530274
## 504 0.796660955 -0.7729186782 -0.9818712 -0.80241764 1.17530274
## 505 0.736267749 -0.6677760011 -0.9818712 -0.80241764 1.17530274
## 506 0.434301716 -0.6126402069 -0.9818712 -0.80241764 1.17530274
## black lstat medv
## 1 0.440615895 -1.074498970 0.159527789
## 2 0.440615895 -0.491952525 -0.101423917
## 3 0.396035074 -1.207532413 1.322937477
## 4 0.415751408 -1.360170785 1.181588636
## 5 0.440615895 -1.025486649 1.486032293
## 6 0.410165113 -1.042290874 0.670558212
## 7 0.426376321 -0.031236706 0.039924923
## 8 0.440615895 0.909799859 0.496590409
## 9 0.328123258 2.419379350 -0.655946292
## 10 0.328999540 0.622727693 -0.394994586
## 11 0.392639483 1.091845624 -0.819041108
## 12 0.440615895 0.086392864 -0.394994586
## 13 0.370513376 0.428078760 -0.090550930
## 14 0.440615895 -0.615183504 -0.231899770
## 15 0.255720500 -0.335113097 -0.471105500
## 16 0.426595391 -0.585776111 -0.286264709
## 17 0.330533033 -0.850442645 0.061670899
## 18 0.329437681 0.282442149 -0.547216415
## 19 -0.741378303 -0.134862757 -0.253645746
## 20 0.375442459 -0.192277190 -0.471105500
## 21 0.217930861 1.171665689 -0.971262937
## 22 0.392749018 0.164812578 -0.318883672
## 23 0.440615895 0.849584722 -0.797295133
## 24 0.414765591 1.012025558 -0.873406047
## 25 0.412465352 0.510699530 -0.753803182
## 26 -0.583319029 0.540106923 -0.938643973
## 27 0.221326452 0.302047077 -0.645073304
## 28 -0.550896614 0.647934029 -0.840787084
## 29 0.342472368 0.020576319 -0.449359525
## 30 0.258020739 -0.094252548 -0.166661844
## 31 0.038293155 1.392921311 -1.069119826
## 32 0.219683424 0.054184768 -0.873406047
## 33 -1.359047220 2.108501199 -1.014754888
## 34 0.022958229 0.797771697 -1.025627875
## 35 -1.186967442 1.076441751 -0.982135924
## 36 0.440615895 -0.416333515 -0.394994586
## 37 0.228774844 -0.174072614 -0.275391721
## 38 0.440615895 -0.543765550 -0.166661844
## 39 0.402607185 -0.353317674 0.235638703
## 40 0.426704926 -1.166922204 0.898890955
## 41 0.426595391 -1.494604580 1.344683452
## 42 0.314759966 -1.094103899 0.442225470
## 43 0.292414788 -0.958269752 0.300876629
## 44 0.413889309 -0.730012370 0.235638703
## 45 0.358354970 -0.434538092 -0.144915868
## 46 0.440615895 -0.342114857 -0.351502635
## 47 0.440615895 0.209623843 -0.275391721
## 48 0.395049257 0.860787538 -0.645073304
## 49 0.440615895 2.542610329 -0.884279035
## 50 0.440615895 0.496696010 -0.340629648
## 51 0.425938180 0.111599201 -0.308010684
## 52 0.408522085 -0.451342316 -0.221026782
## 53 0.440615895 -1.032488409 0.268257666
## 54 0.440615895 -0.591377519 0.094289862
## 55 0.440615895 0.300646725 -0.394994586
## 56 0.429990982 -1.098304955 1.399048391
## 57 0.440615895 -0.963871160 0.235638703
## 58 0.396801820 -1.218735230 0.985874857
## 59 0.372485009 -0.811232788 0.083416874
## 60 0.440615895 -0.480749709 -0.318883672
## 61 0.421009097 0.069588640 -0.416740562
## 62 0.234470674 0.250234052 -0.710311231
## 63 0.440615895 -0.829437365 -0.036185991
## 64 0.426157250 -0.441539852 0.268257666
## 65 0.400526017 -0.644590896 1.138096685
## 66 0.440615895 -1.117909883 0.105162850
## 67 0.440615895 -0.337913801 -0.340629648
## 68 0.433057967 -0.637589136 -0.057931966
## 69 0.440615895 0.061186528 -0.558089402
## 70 0.440615895 -0.540964846 -0.177534831
## 71 0.296358054 -0.830837717 0.181273764
## 72 0.221983663 -0.388326475 -0.090550930
## 73 0.375004318 -0.998879961 0.029051936
## 74 0.224502972 -0.716008850 0.094289862
## 75 0.418927928 -0.822435605 0.170400776
## 76 0.290881295 -0.519959566 -0.123169893
## 77 0.186056121 -0.095652900 -0.275391721
## 78 0.331737920 -0.333712745 -0.188407819
## 79 0.325603949 -0.043839875 -0.144915868
## 80 0.431414939 -0.497553933 -0.242772758
## 81 0.440615895 -1.031088057 0.594447298
## 82 0.426704926 -0.760820115 0.148654801
## 83 0.440615895 -0.830837717 0.246511690
## 84 0.372046868 -0.720209906 0.039924923
## 85 0.440615895 -0.424735627 0.148654801
## 86 0.390229709 -0.857444405 0.442225470
## 87 0.430648193 0.028978431 -0.003567028
## 88 0.421447237 -0.589977167 -0.036185991
## 89 0.440615895 -1.001680665 0.116035838
## 90 0.431414939 -0.973673624 0.670558212
## 91 0.388915287 -0.538164142 0.007305960
## 92 0.403921608 -0.623585616 -0.057931966
## 93 0.419913745 -0.629187024 0.039924923
## 94 0.434372389 -0.902255670 0.268257666
## 95 0.440615895 -0.288901480 -0.210153795
## 96 0.014304949 -0.840640181 0.637939249
## 97 0.385081555 -0.183875078 -0.123169893
## 98 0.440615895 -1.182326077 1.757856987
## 99 0.403702537 -1.271948607 2.312379361
## 100 0.440615895 -0.905056374 1.159842661
## 101 0.417175365 -0.452742668 0.540082359
## 102 0.426157250 -0.697804274 0.431352482
## 103 -3.131326538 -0.283300072 -0.427613550
## 104 0.413998845 0.110198849 -0.351502635
## 105 0.394501581 -0.045240227 -0.264518733
## 106 0.409398367 0.534505515 -0.329756660
## 107 0.427143067 0.841182610 -0.329756660
## 108 0.339733988 0.201221731 -0.231899770
## 109 0.422433054 -0.053642339 -0.297137697
## 110 0.378509444 0.405673128 -0.340629648
## 111 0.403264396 0.048583360 -0.090550930
## 112 0.426266786 -0.349116618 0.029051936
## 113 0.419256534 0.498096362 -0.405867574
## 114 0.440615895 0.621327341 -0.416740562
## 115 0.351235183 -0.308506409 -0.438486537
## 116 -0.128857540 0.435080520 -0.460232513
## 117 0.401183228 -0.085850436 -0.144915868
## 118 0.414436985 -0.329511689 -0.362375623
## 119 -0.197645637 0.380466791 -0.231899770
## 120 0.381466894 0.134004834 -0.351502635
## 121 0.355726125 0.240431588 -0.057931966
## 122 0.229979731 0.226428068 -0.242772758
## 123 0.234580209 0.738956911 -0.221026782
## 124 0.149361834 1.786420232 -0.568962390
## 125 0.248710248 0.689944590 -0.405867574
## 126 0.310488093 0.302047077 -0.123169893
## 127 0.028654058 2.045485358 -0.742930194
## 128 0.388148541 0.635330861 -0.688565255
## 129 0.440615895 0.383267495 -0.492851476
## 130 0.440615895 0.796371345 -0.895152022
## 131 0.420242350 -0.007430722 -0.362375623
## 132 0.440615895 -0.055042691 -0.318883672
## 133 0.318593697 -0.214682823 0.050797911
## 134 0.350687507 0.332854822 -0.449359525
## 135 -1.028689097 0.652135085 -0.753803182
## 136 0.416189548 0.603122764 -0.481978488
## 137 0.236332772 0.594720652 -0.558089402
## 138 0.409726972 0.271239333 -0.590708366
## 139 0.387381794 1.213676250 -1.003881900
## 140 0.440615895 0.813175569 -0.514597451
## 141 0.344005860 1.611376228 -0.927770986
## 142 0.440615895 3.046737061 -0.884279035
## 143 0.440615895 1.983869868 -0.993008912
## 144 0.440615895 1.927855787 -0.753803182
## 145 0.440615895 2.329756820 -1.166976716
## 146 -2.012862748 2.121104367 -0.949516961
## 147 -2.052733556 0.559711851 -0.753803182
## 148 0.383767133 2.363365269 -0.862533059
## 149 0.003460966 2.193922673 -0.514597451
## 150 -0.052840120 1.231880827 -0.775549157
## 151 0.176636095 0.202622083 -0.112296905
## 152 -0.165113687 0.087793216 -0.318883672
## 153 -0.146711775 -0.074647619 -0.786422145
## 154 -1.037561447 0.439281577 -0.340629648
## 155 -0.390537100 0.345457990 -0.601581353
## 156 -2.942816482 0.331454470 -0.753803182
## 157 -2.936025300 0.488293898 -1.025627875
## 158 0.074001626 -1.129112700 2.040554668
## 159 -0.030494942 -0.871447926 0.192146752
## 160 0.083640722 -0.737014131 0.083416874
## 161 -0.194469117 -1.001680665 0.485717421
## 162 0.194490331 -1.529613381 2.986504601
## 163 0.360764744 -1.503006692 2.986504601
## 164 0.348058662 -1.306957408 2.986504601
## 165 0.421009097 -0.141864517 0.018178948
## 166 -1.276238619 -0.398128939 0.268257666
## 167 0.138298780 -1.253744030 2.986504601
## 168 -1.413705278 -0.071846915 0.137781813
## 169 -0.652654802 -0.217483527 0.137781813
## 170 -0.291736362 -0.186675782 -0.025313003
## 171 -0.705231691 0.248833700 -0.558089402
## 172 -0.093587210 -0.087250788 -0.373248611
## 173 0.440615895 0.285242853 0.061670899
## 174 0.425280969 -0.505956045 0.116035838
## 175 0.400416482 -0.421934923 0.007305960
## 176 0.375551994 -1.025486649 0.746669127
## 177 0.400416482 -0.356118378 0.072543887
## 178 0.426376321 -0.891052854 0.224765715
## 179 0.378947585 -0.802830676 0.801034065
## 180 0.440615895 -1.066096858 1.594762170
## 181 0.425938180 -0.713208146 1.877459852
## 182 0.440615895 -0.448541612 1.486032293
## 183 0.410165113 -1.096904603 1.670873085
## 184 0.440615895 -0.976474328 1.083731747
## 185 0.375990135 0.185817859 0.420479494
## 186 0.333380947 0.069588640 0.768415102
## 187 0.393844370 -1.148717628 2.986504601
## 188 0.407426733 -0.836439125 1.029366808
## 189 0.286609423 -1.133313756 0.790161078
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## 503 0.440615895 -0.500354637 -0.210153795
## 504 0.440615895 -0.982075736 0.148654801
## 505 0.402826256 -0.864446166 -0.057931966
## 506 0.440615895 -0.668396881 -1.156103728boston_scaled<-data.frame(boston_scaled, crime)Divide the dataset into train and test
n<-nrow(boston_scaled)
ind<-sample(n, size = n*0.8)
train<-boston_scaled[ind, ]
test<-boston_scaled[-ind, ]
correct_classes<-test$crimeLinear Discriminant Analysis
lda.fit<-lda(crime~., data=train)
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2623762 0.2549505 0.2301980 0.2524752
##
## Group means:
## zn indus chas nox rm
## low 0.9724027 -0.9417505 -0.123759247 -0.8833222 0.4306105
## med_low -0.1560872 -0.2347076 -0.004759149 -0.5485858 -0.1737656
## med_high -0.3563036 0.1249185 0.108680633 0.2809191 0.2250725
## high -0.4872402 1.0171096 -0.040734936 1.0370219 -0.3903153
## age dis rad tax ptratio
## low -0.8921034 0.9075613 -0.6882537 -0.7457144 -0.420885405
## med_low -0.2658939 0.3091336 -0.5581649 -0.4607568 0.003946553
## med_high 0.4027872 -0.3522652 -0.4014631 -0.3258307 -0.292861051
## high 0.8108363 -0.8626504 1.6382099 1.5141140 0.780871766
## black lstat medv
## low 0.3753774 -0.75738529 0.53064543
## med_low 0.3096863 -0.06205805 -0.05117594
## med_high 0.1571859 -0.07448199 0.28953599
## high -0.7872832 0.91360278 -0.69730628
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.121325608 0.77207565 -0.75846084
## indus 0.053895373 -0.41181402 0.19835527
## chas -0.008812182 -0.01759874 0.14678878
## nox 0.520596914 -0.57053926 -1.46183169
## rm 0.018117723 -0.06453497 -0.18151489
## age 0.197398037 -0.35101243 -0.06474741
## dis -0.074070315 -0.31204251 0.09798344
## rad 3.149134426 0.90341071 -0.05691434
## tax -0.049102223 0.02712580 0.60007210
## ptratio 0.090762409 0.08925563 -0.17598439
## black -0.119527655 -0.00867947 0.10688625
## lstat 0.221627662 -0.27198749 0.29787702
## medv 0.090152036 -0.39838427 -0.30917956
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9497 0.0363 0.0140lda.arrows<-function(x, myscale=1, arrow_heads= 0.1, color="red", tex=0.75, choices =c (1,2)){
heads<-coef(x)
arrows(x0 = 0, y0=0, x1 = myscale*heads[,choices[1]],y1=myscale*heads[,choices[2]], col=color, length= arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}Convert target classes to numeric data
classes<-as.numeric(train$crime)Plot the lda results
plot(lda.fit, dimen=2, col=classes, pch=classes)
lda.arrows(lda.fit, myscale=1)
Predict the classes with the LDA model and Cross-tabulate the results
lda.pred<-predict(lda.fit, newdata = test)
table(correct=correct_classes, predict=lda.pred$class)## predict
## correct low med_low med_high high
## low 13 7 1 0
## med_low 9 13 1 0
## med_high 0 14 18 1
## high 0 0 0 25Distance measures
library(MASS)
data("Boston")
boston_scaled<-scale(Boston)
boston_scaled<-as.data.frame(boston_scaled)Euclidean distane matrix
dist_eu<-dist(boston_scaled)
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970K-means
km<-kmeans(boston_scaled, centers=3)
pairs(boston_scaled, col=km$cluster)
Investigate the optimal number of clusters
set.seed(11)
k_max<-20
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
qqplot(x=1:k_max, y = twcss, geom="Line")## Warning in plot.window(...): "geom" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "geom" is not a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "geom" is not
## a graphical parameter
## Warning in axis(side = side, at = at, labels = labels, ...): "geom" is not
## a graphical parameter## Warning in box(...): "geom" is not a graphical parameter## Warning in title(...): "geom" is not a graphical parameter
Perform k-means with different number of clusters
km<-kmeans(boston_scaled, centers=6)
pairs(boston_scaled, col = km$cluster)
library(GGally)## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2##
## Attaching package: 'GGally'## The following object is masked from 'package:dplyr':
##
## nasaggpairs(boston_scaled)
Super Bonus
model_predictors<- dplyr::select(train, -crime)Check th dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3Matrix multiplication
matrix_product <- as.matrix(model_predictors)%*%lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)Create a 3D plot
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutplot_ly(x=matrix_product$LD1, y=matrix_product$LD2, z=matrix_product$LD3, type="scatter3d", mode="markers")plot_ly(x=matrix_product$LD1, y=matrix_product$LD2, z=matrix_product$LD3, type="scatter3d", mode="markers", color = train$crime)km <-kmeans(boston_scaled, centers = 6)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = km$cluster[ind])